assumptions about behavior that have been well documented empirically.One may view our paper as a companion paper of Allen and Gale.They study the possibility of price manipulation under rational expectations with information asymmetry while we provide a case of market manipulation under behavioral bias and limits to arbitrage but with no fundamental risk or information asymmetry. 2.The Model Economy We consider a discrete-time market in which there exist a speculative asset and a riskfree bond.The riskfree bond yields a zero net return each period of time.There are three classes of investors,a manipulator,arbitrageurs,and behavior-driven traders,who buy and sell the speculative asset following their own rules.The characteristics of these investors are described in detail in the following assumptions. Assumption 1.We consider a discrete-time economy that begins at time t=0,and ends at time t=T(namely,t=0,1,2,..,T).A continuum number of new behavior-driven investors,with measure 1,enter the market at the beginning of each period t.They are price-takers and each of them has a probability of q to buy a share of the speculative asset if the price of the asset at time t0,P,is greater than the asset price at time t1, P..If p<P each new behavior-driven investor has a probability of q2 to buy a share of the speculative asset,where q2<q. At the beginning of the economy,t=0,the price of the speculative asset (P)is equal to the fundamental value of the asset,and the behavior-driven investors are endowed with 10
10 assumptions about behavior that have been well documented empirically. One may view our paper as a companion paper of Allen and Gale. They study the possibility of price manipulation under rational expectations with information asymmetry while we provide a case of market manipulation under behavioral bias and limits to arbitrage but with no fundamental risk or information asymmetry. 2. The Model Economy We consider a discrete-time market in which there exist a speculative asset and a riskfree bond. The riskfree bond yields a zero net return each period of time. There are three classes of investors, a manipulator, arbitrageurs, and behavior-driven traders, who buy and sell the speculative asset following their own rules. The characteristics of these investors are described in detail in the following assumptions. Assumption 1. We consider a discrete-time economy that begins at time t = 0, and ends at time t = T (namely, t = 0,1,2,L,T ). A continuum number of new behavior-driven investors, with measure 1, enter the market at the beginning of each period t. They are price-takers and each of them has a probability of q1 to buy a share of the speculative asset if the price of the asset at time t>0, Pt , is greater than the asset price at time t-1, Pt -1 . If Pt £ Pt -1 , each new behavior-driven investor has a probability of q2 to buy a share of the speculative asset, where q2 < q1 . At the beginning of the economy, t = 0, the price of the speculative asset ( P0 ) is equal to the fundamental value of the asset, and the behavior-driven investors are endowed with
q shares of the speculative asset in total.Those investors who own the speculative asset at the beginning of the economy take Po as the initial acquiring cost per share of the speculative asset. The new behavior-driven investors at time t>0 who do not buy the speculative asset choose to leave the market right away.The old generations of behavior-driven investors who entered the market before t>0 do not buy any more shares at time t.Behavior-driven investors like to take quick profits.They sell their shares as soon as they have made a profit and then leave the market.Consider a behavior-driven investor who buys a share of the speculative asset at time t and has not sold his share by the beginning of time t+k (k>0).If P<Pk,he shall liquidate his share in the period of t+k for sure; if PzP,he will have a probability of q3<I to liquidate his share in the period of t+k.Behavior-driven investors leave the market right after they have liquidated their shares. This assumption is made on the basis of two important empirical observations:trend chasing (momentum trading/positive feedback trading)and dispositional effect. In an original article,Jegadeesh and Titman (1993)find that the winners of the stock market over past several months tend to outperform in the next several months as well. This phenomenon is now termed as momentum and has been well documented in the behavioral finance literature.In the BSV model,momentum can occur because of the investors'conservatism.Hong and Stein (1999),as introduced earlier,explicitly add momentum traders-traders buying stocks after a price increase-to their model.Many other researchers,including DeLong,Shleifer,Summers,and Waldermann (1990)and Cutler,Summers,and Poterba (1990),have also investigated momentum trading or 11
11 q1 shares of the speculative asset in total. Those investors who own the speculative asset at the beginning of the economy take P0 as the initial acquiring cost per share of the speculative asset. The new behavior-driven investors at time t>0 who do not buy the speculative asset choose to leave the market right away. The old generations of behavior-driven investors who entered the market before t>0 do not buy any more shares at time t. Behavior-driven investors like to take quick profits. They sell their shares as soon as they have made a profit and then leave the market. Consider a behavior-driven investor who buys a share of the speculative asset at time t and has not sold his share by the beginning of time t + k (k > 0) . If Pt < Pt+k , he shall liquidate his share in the period of t + k for sure; if Pt ³ Pt+k , he will have a probability of 1 q3 < to liquidate his share in the period of t + k . Behavior-driven investors leave the market right after they have liquidated their shares. This assumption is made on the basis of two important empirical observations: trend chasing (momentum trading/positive feedback trading) and dispositional effect. In an original article, Jegadeesh and Titman (1993) find that the winners of the stock market over past several months tend to outperform in the next several months as well. This phenomenon is now termed as momentum and has been well documented in the behavioral finance literature. In the BSV model, momentum can occur because of the investors’ conservatism. Hong and Stein (1999), as introduced earlier, explicitly add momentum traders— traders buying stocks after a price increase— to their model. Many other researchers, including DeLong, Shleifer, Summers, and Waldermann (1990) and Cutler, Summers, and Poterba (1990), have also investigated momentum trading or
positive feedback trading.The simplest way of motivating positive feedback trading is extrapolative expectations.Namely,as investors form expectations by extrapolating trends,they buy into price trends.This can be due to some importart psychological biases of investors,including representativeness and the law of small numbers (Barberis and Thaler,2003). Robert J.Shiller(2002,p14)has the following vivid description on momentum trading or feedback trading: When speculative prices go up,creating successes of some investors,this may attract public attention,promote word-of-mouth enthusiasm,and heighten expectations for further price increases....This process in turn increases investor demand,and thus generates another round of price increases....The high prices are ultimately not sustainable,since they are high only because of expectations of further price increases.... The story about tulip mania in Holland in the 1630s provides us a real example on momentum trading with little fundamental news (Charles MacKay (1841,pp 118-119)or Shiller (2002,p15)): Many individuals grew suddenly rich.A golden belt hung temptingly out before the people,and one after another,they rushed to the tulip marts,like flies around a honey pot....At last,however,the more prudent began to see that this folly could not last forever.Rich people no longer bought the flowers to keep them in their gardens,but to sell them again at cent per cent profit.It was seen that somebody must lose fearfully in the end.As this conviction spread,prices fell,and never rose again. 12
12 positive feedback trading. The simplest way of motivating positive feedback trading is extrapolative expectations. Namely, as investors form expectations by extrapolating trends, they buy into price trends. This can be due to some important psychological biases of investors, including representativeness and the law of small numbers (Barberis and Thaler, 2003). Robert J. Shiller (2002, p14) has the following vivid description on momentum trading or feedback trading: When speculative prices go up, creating successes of some investors, this may attract public attention, promote word-of-mouth enthusiasm, and heighten expectations for further price increases. … This process in turn increases investor demand, and thus generates another round of price increases. … The high prices are ultimately not sustainable, since they are high only because of expectations of further price increases. … The story about tulip mania in Holland in the 1630s provides us a real example on momentum trading with little fundamental news (Charles MacKay (1841, pp 118-119) or Shiller (2002, p15)): Many individuals grew suddenly rich. A golden belt hung temptingly out before the people, and one after another, they rushed to the tulip marts, like flies around a honey pot. … At last, however, the more prudent began to see that this folly could not last forever. Rich people no longer bought the flowers to keep them in their gardens, but to sell them again at cent per cent profit. It was seen that somebody must lose fearfully in the end. As this conviction spread, prices fell, and never rose again
Dispositional effect is another well-documented empirical phenomenon.According to Shefrin and Statman (1985),Odean (1998),Grinblatt and Han (2001),etc.,investors, especially the individual ones,are more likely to sell stocks that have gone up in value relative to their purchase price,rather than stocks that have gone down.Two behavioral explanations for the dispositional effect have been suggested in the literature.The first explanation suggests that investors may have a biased belief in mean-reversion.The second explanation relies on prospect theory and narrow framing. Assumption 2.There is a manipulator in the market who is a large market player and is able to influence the asset price.In other words,the manipulator is a price-setter rather than a price-taker.He enters the market at time I without any initial endowment of the speculative asset.At each period of time tz1,the manipulator sets a price target for that period and then submits his order to clear the market at the target price. The assumption that the manipulator is a large trader is conventional in the literature on trade-based manipulation.In order to move the market with strategic trading,the manipulator must have the power to influence the price(see Jarrow(1992)and Allen and Gale (1992)).In real markets,many investors,such as investment institutions,wealthy individuals,or a group of investors (e.g.,trading pool in the US history)can be classified as large traders.These traders often not only have deep pockets,but also are professional and influential in the securities markets. Assumption 3.There is also a continuum number of arbitrageurs,with measure 1,enters the market at time t=1.They are price-takers and trade shares of the speculative asset based on recent price movements.If the price moves up in the current period,they sell some shares to take profits.If the price goes down,they buy.Formally,they submit the following orders at time t: 13
13 Dispositional effect is another well-documented empirical phenomenon. According to Shefrin and Statman (1985), Odean (1998), Grinblatt and Han (2001), etc., investors, especially the individual ones, are more likely to sell stocks that have gone up in value relative to their purchase price, rather than stocks that have gone down. Two behavioral explanations for the dispositional effect have been suggested in the literature. The first explanation suggests that investors may have a biased belief in mean-reversion. The second explanation relies on prospect theory and narrow framing. Assumption 2. There is a manipulator in the market who is a large market player and is able to influence the asset price. In other words, the manipulator is a price-setter rather than a price-taker. He enters the market at time 1 without any initial endowment of the speculative asset. At each period of time t ³ 1, the manipulator sets a price target for that period and then submits his order to clear the market at the target price. The assumption that the manipulator is a large trader is conventional in the literature on trade-based manipulation. In order to move the market with strategic trading, the manipulator must have the power to influence the price (see Jarrow (1992) and Allen and Gale (1992)). In real markets, many investors, such as investment institutions, wealthy individuals, or a group of investors (e.g., trading pool in the US history) can be classified as large traders. These traders often not only have deep pockets, but also are professional and influential in the securities markets. Assumption 3. There is also a continuum number of arbitrageurs, with measure 1, enters the market at time t=1. They are price-takers and trade shares of the speculative asset based on recent price movements. If the price moves up in the current period, they sell some shares to take profits. If the price goes down, they buy. Formally, they submit the following orders at time t:
D,=-a(e-P)=-(AP） 0 where a-0. Although the new trades of the arbitrageurs in each period only depend on short term price movements,the total position of the speculative asset held by the arbitrageurs,O,, is negatively proportional to price deviation from fundamentals.This is because the rhaveed of-2-ad-P.）上-uP-R） shares of the speculative asset at time t-1,if they buy additional -a(p-P_)shares at time t,the total position of the speculative asset held by them will be =(P-P)shares.The arbitrageurs play two roles in our model.First,they provide necessary liquidity to the market so that trading can take place at equilibrium for each period.For instance,if the manipulator wants to move the asset price up by submitting a purchasing order,there must be some investors selling sufficient number of shares of the speculative asset.Because the behavior-driven investors in a sense are momentum followers,a new class of investors is therefore needed in the model.Second, our model rules out fundamental risk.The arbitrageurs'trading strategy ensures that the price of speculative asset will not move away from fundamentals explosively.We call a arbitrage parameter and will discuss its meaning and implication further in the next section Assumption 4.Although the manipulator enters the market at time 1,the market already existed at time 0.The price of the speculative asset at time 0 was Po,which was equal to the fundamental value of the asset.There were q behavior-driven investors who held one share of the speculative asset per person at the market close ofday 0. 14
14 ( ) ( ) Da,t = -a Pt - Pt -1 = -a DPt (1) where a>0. Although the new trades of the arbitrageurs in each period only depend on short term price movements, the total position of the speculative asset held by the arbitrageurs, Qt , is negatively proportional to price deviation from fundamentals. This is because the arbitrageurs have already held a portfolio of ( ) ( ) 1 0 1 1 Q 1 P P 1 Pt P t j t = - j - j = - - - - = - å a - a shares of the speculative asset at time t -1, if they buy additional ( ) - a Pt - Pt -1 shares at time t , the total position of the speculative asset held by them will be ( ) Qt = -a Pt - P0 shares. The arbitrageurs play two roles in our model. First, they provide necessary liquidity to the market so that trading can take place at equilibrium for each period. For instance, if the manipulator wants to move the asset price up by submitting a purchasing order, there must be some investors selling sufficient number of shares of the speculative asset. Because the behavior-driven investors in a sense are momentum followers, a new class of investors is therefore needed in the model. Second, our model rules out fundamental risk. The arbitrageurs’ trading strategy ensures that the price of speculative asset will not move away from fundamentals explosively. We call a arbitrage parameter and will discuss its meaning and implication further in the next section. Assumption 4. Although the manipulator enters the market at time 1, the market already existed at time 0. The price of the speculative asset at time 0 was P0 , which was equal to the fundamental value of the asset. There were q1 behavior-driven investors who held one share of the speculative asset per person at the market close of day 0